Physics 351, Spring 2017, Homework #12. Due at start of class, Friday, April 14, 2017 Course info is at positron.hep.upenn.edu/p351 When you finish this homework, remember to visit the feedback page at positron.hep.upenn.edu/q351 to tell me how the homework went for you. 1. [Here s a problem that appeared on a midterm exam 3 or 4 years ago; the last two parts were extra-credit on the exam, but you have to solve them!] Consider the double pendulum consisting of two bobs confined to move in a plane. The rods are of equal length l, and the bobs have equal mass m. The generalized coordinates used to describe the system are θ 1 and θ 2, the angles that the rods make with the vertical (see left figure below). (a) Write the Lagrangian for the system. (This could be an opportunity to practice writing (v 1 +v 2 ) 2 = v 2 1 +v 2 2 +2v 1 v 2.) (b) Next, simplify your Lagrangian from part (a) by assuming that angles θ 1 and θ 2 are both small. Keep terms up to second order in the angles, the angular velocities, and their products. (c) Find the two Lagrange equations of motion, which will be a set of coupled, linear differential equations. (d) Solve the equations of motion (e.g. using the techniques of Chapter 11). 2. [Here s another problem from a midterm exam 3 or 4 years ago; the last part was extra-credit on the exam.] A block of mass M moves on a frictionless horizontal rail. A pendulum of length L and mass m hangs from the block. (See right figure above.) Let x be the displacement of the block, and let θ be the angular displacement of the pendulum w.r.t. the vertical. (a) Write the Lagrangian for the system. (Another possible opportunity to write (v 1 + v 2 ) 2 = v 2 1 + v 2 2 + 2v 1 v 2?) (b) Which of the coordinates is ignorable ( cyclic )? What is the associated conserved quantity? This is an example of what conservation law? (c) Find the Lagrange equations of motion for the system. (d) Simplify the equations of motion found in part (c) for the case phys351/hw12.tex page 1 of 6 2017-04-06 20:16
of small oscillations (where you can discard any terms of second order or higher in the displacements, velocities, or their products). (e) Solve the system of differential equations from part (d) and determine the most general motion of the system. (Your solution should have four arbitrary constants. Using the results of part (b) should help you to simplify the problem.) 3. Hamiltonian treatment of the symmetric top. [Here s a problem that appeared on a previous year s final exam.] Consider a symmetric top (λ 1 = λ 2 ) whose tip has a fixed location in space. Using the Euler angles φ, θ, and ψ (whose detailed definitions are not needed for you to solve this problem) to represent the top s orientation, the top s Lagrangian can be written as L = 1 2 λ 1 φ 2 sin 2 θ + 1 2 λ 1 θ 2 + 1 2 λ 3( ψ + φ cos θ) 2 MgR cos θ where M is the mass of the top and R is the distance from the contact point to the top s CoM. λ 3 is the moment of inertia for the top s symmetry axis, and λ 1 is the moment of inertia for the other two principal axes. (a) Calculate the three generalized momenta, p φ, p θ, and p ψ. (b) The simplest way to construct the Hamiltonian is to realize that the coordinates are natural, so H = T + U. Use this to show that the Hamiltonian is given by H = (p φ p ψ cos θ) 2 2λ 1 sin 2 θ + p2 θ 2λ 1 + p2 ψ 2λ 3 + MgR cos θ (c) Two of the Euler-angle coordinates are ignorable. Which ones? The corresponding generalized momenta are constant. Use this to show that the Hamiltonian can be written as H = p2 θ 2λ 1 + U eff (θ) What is the effective potential energy U eff for this system? 4. Two masses m 1 and m 2 are joined by a massless spring (force constant k and natural length l 0 ) and are confined to move in a frictionless horizontal plane, with CM and relative positions R and r as defined in 8.2. (a) Write down the Hamiltonian H using as generalized coordinates X, Y, r, φ, where (X, Y ) are the rectangular components of R, and (r, φ) are the polar coordinates of r. Which coordinates are ignorable and which are not? Explain. (b) Write down the 8 Hamilton equations of motion. (c) Solve the r equations for the special case that p φ = 0 and describe the motion. 5. When you spin a coin around a vertical diameter on a table, it will lose energy and go into a wobbling motion, whose frequency increases as the coin s angle w.r.t. horizontal decreases. Consider the moment when the coin makes an angle θ w.r.t. phys351/hw12.tex page 2 of 6 2017-04-06 20:16
the horizontal surface of the table. Assume that the CM of the coin is motionless and that the contact point moves along a circle on the table, as shown in the left figure below. Let the radius of the coin be R, and let Ω be the angular velocity of the motion of the contact point. Assume that the coin rolls without slipping. (a) Show that the angular velocity of the coin is ω = Ω sin θ ê 1, where ê 1 points upward along the coin, diametrically away from the contact point. (b) Show that Ω = 2 g/(r sin θ). 6. Consider the modified Atwood machine shown in the right figure above. The two weights on the left have equal masses m and are connected by a massless spring of Hooke s-law constant k. The weight on the right has mass M = 2m, and the pulley is massless and frictionless. The coordinate x is the extension of the spring from its equilibrium length; that is, the length of the spring is l e + x, where l e is the equilibrium length (with all the weights in position and M held stationary). (a) Show that the total potential energy (spring plus gravitational) is just U = 1 2 kx2 (plus a constant that we can take to be zero). (b) Find the two momenta conjugate to x and y. Solve for ẋ and ẏ, and write down the Hamiltonian. Show that the coordinate y is ignorable. (c) Write down the four Hamilton equations an solve them for the following initial conditions: You hold the mass M fixed with the whole system in equilibrium and y = y 0. Still holding M fixed, you pull the lower mass m down a distance x 0, and at t = 0 you let go of both masses. [Hint: Write down the initial values of x, y, and their momenta. You can solve the x equations by combining them into a second-order equation for x. Once you know x(t), you can quickly write down the other three variables.] Describe the motion. In particular, find the frequency with which x oscillates. 7. [This problem is adapted from a problem on the final exam I took for the analogous course in fall 1990.] A uniform, infinitesimally thick, square plate of mass m and side phys351/hw12.tex page 3 of 6 2017-04-06 20:16
length d is allowed to undergo torque-free rotation. At time t = 0, the normal to the plate, ê 3, is aligned with ẑ, but the angular velocity vector ω deviates from ẑ by a small angle α. The figure below depicts the situation at time t = 0, at which time ê 1 = ˆx, ê 2 = ŷ, ê 3 = ẑ, and ω = ω(cos αẑ + sin αˆx). (a) Show that the inertia tensor has the form I = I 0 1 0 0 0 1 0 0 0 2 and find the constant I 0. (b) Calculate the angular momentum vector L at t = 0. (c) Draw a sketch showing the vectors ê 3, ω, and L at t = 0. Be sure that the relative orientation of L and ω makes sense. This relative orientation is different for frisbee-like ( oblate ) objects (λ 3 > λ 1 ) than it is for the American-football-like ( prolate ) object (λ 3 < λ 1 ) drawn on Taylor s page 400. (d) Draw and label the body cone and the space cone on your sketch. (e) Calculate the precession frequencies Ω body and Ω space. Indicate the directions of the precession vectors Ω body and Ω space on your drawing. (You puzzled through these directions when you solved problem 10.46.) (f) You argued in HW11 that Ω space = Ω body + ω. Verify (by writing out components) that this relationship holds for the Ω space and Ω body that you calculate for t = 0. (g) Find the maximum angle between ẑ and ê 3 during subsequent motion of the plate. Show that in the limit α 1, this maximum angle equals α (dropping terms O(α 2 ) and higher). (h) When is this maximum deviation first reached? (i) As a check, verify (for the α 1 limit) that Feynman indeed misremembered which way the factor of two had gone in this anecdote about a plate tossed through the air in a Cornell cafeteria: positron.hep.upenn.edu/p351/feynman Remember online feedback at positron.hep.upenn.edu/q351 phys351/hw12.tex page 4 of 6 2017-04-06 20:16
XC00*. Optional/extra-credit. If there are extra-credit problems from earlier homework assignments that you didn t have time to do sooner, you can feel free to turn them in with HW12 for full credit. Just clearly indicate which problem you re solving. XC1. Optional/extra-credit. Consider a rotating reference frame such as a frame fixed on the earth s surface. A particle is thrown vertically up with initial speed v 0, reaches a maximum height, and falls back to the ground. Show that the Coriolis deflection when it reaches the ground is four times as large as and in the opposite direction from the Coriolis deflection when it is dropped from rest at the same maximum height. Can you explain why? XC2. Optional/extra-credit. Assume that a piece of toast is a rigid uniform square of side length l. You butter the toast and then drop it from a height H above a table; the table is a height h above the floor. The toast starts off parallel to the table, and as it falls, it clips the edge of the table and collides elastically, causing the toast to start to rotate. You want to find the value of H, in terms of h and l, that leads to the sad situation in which the toast makes exactly one-half revolution and lands on the floor butter-side-down. Show that H = π2 l 2. [Hint: with a clever 6 (6h πl) choice of origin, you can argue that the angular momentum of the toast is conserved during the collision with the table.] XC3. Optional/extra-credit. (a) A small ball of radius r and uniform density rolls without slipping at the bottom of a fixed cylinder of radius R r. Show that 5g the frequency of small oscillations is ω =. [You ll need I = 2 7R 5 Mr2 for a uniform sphere about an axis through its center.] (b) Generalize your result for the case where the sphere s density is not uniform (but is still spherically symmetric), so its moment of inertia is given by I = βmr 2. XC4. Optional/extra-credit. Consider a top made of a wheel with all its mass on the rim. A massless rod (perpendicular to the plane of the wheel) connects the CM to a pivot. Initial conditions have been set up so that the top undergoes precession, with the rod always horizontal. In the language of the figure below (Morin s Fig. 9.30), we may write the angular velocity of the top as ω = Ωẑ + ω ˆx 3 (where ˆx 3 = ê 3 is horizontal here). Consider things in the frame rotating around the ẑ axis with angular speed Ω. In this frame, the top spins with angular speed ω around its fixed symmetry axis. Therefore, in this frame we must have τ = 0, because L is constant. Verify explicitly that τ = 0 (calculated w.r.t. the pivot) in this rotating frame (you will need to find the relation between ω and Ω). In other words, show that the torque due to gravity is exactly canceled by the torque due to the Coriolis force (you can quickly show that the centrifugal force provides no net torque). Remember that phys351/hw12.tex page 5 of 6 2017-04-06 20:16
HW10/XC3 implies a Coriolis torque of magnitude mω Ωr 2. XC5. Optional/extra-credit. All of the examples in Taylor s Chapter 13 and all of the problems (except this one) treat forces that come from a potential energy U(r) [or occasionally U(r, t)]. However, the proof of Hamilton s equations given in 13.3 applies to any system for which Lagrange s equations hold, and this can include forces not derivable from a potential energy. An important example of such a force is the magnetic force on a charged particle. (a) Use the Lagrangian (Eq. 7.103) to show that the Hamiltonian for a charge q in an electromagnetic field is H = (p qa) 2 /(2m)+qV. (This Hamiltonian plays an important role in the quantum mechanics of charged particles.) (b) Show that Hamilton s equations are equivalent to the familiar Lorentz force equation m r = q(e + v B). Remember online feedback at positron.hep.upenn.edu/q351 phys351/hw12.tex page 6 of 6 2017-04-06 20:16